CMI Lecture Series in Mathematics

 The three theorems proved in this book look very different, but they have one feature in common: their statements do not involve p-adic numbers, but their proofs use p-adic methods. And those methods are quite different from each other, which illustrates well the versatility of the p-adic approach to Diophantine problems

The principal objective of this book is to introduce to a reader with modest background the notion of p-adic numbers and relating concepts (complete fields, etc).
However, the specific feature of the book is that it gives a motivated introduction to the topic. It not only give principal definitions and properties but also show how p-adic method can be used to solve difficult problems that can be stated without using the notion of p-adic numbers. In this book the author's use p-adic methods to prove three highly non-trivial theorems: the Hasse-Minkowski local-global principle for quadratic forms, the Skolem-Mahler-Lech finiteness theorem for zeros of linear recurrences, and Sprindzhuk’s irreducibility theorem, a beautiful version of Hilbert’s classical irreducibility theorem. In its present form, this book is accessible to a graduate student, as well as a curious undergraduate.